Set Up Your Notes Evaluating Piecewise and Step Functions
Evaluating Piecewise Functions • Piecewise functions are functions defined by at least two equations, each of which applies to a different part of the domain • A piecewise function looks like this: Domain restrictions Equations
Evaluating Piecewise Functions • Steps to Evaluate Piecewise Functions • Look at the domain to see which equation to use • Plug in x-value • Solve! Ex. #1: Find a. g(-2) and b. g(2)
Evaluating Piecewise Functions Ex. #2 Which equation would we use to find; g(-5)? g(-2)? g(1)?
Step Functions Looks like a flight of stairs An example of a step function: Graphically, the equation would look like this:
Classwork/Homework • Evaluating WS
Characteristics of Piecewise Functions
Domain and Range of Piecewise Functions • Domain (x): the set of all input numbers - will not include points where the function(s) do not exist. The domain also controls which part of the piecewise function will be used over certain values of x. • Range (y): the set of all outputs.
Points of Discontinuity • These are the points where the function either “jumps” up or down or where the function has a “hole”. • For example, in a previous example Has a point of discontinuity at x=1 The step function also has points of discontinuity at x=1, x=2 and x=3.
Axis of Symmetry • The vertical line that splits the equation in half. This ‘axis of symmetry’ can be found by identifying the x-coordinate of the vertex (h,k), so the equation for the axis of symmetry would be x = h. For the equation the axis of symmetry is located at x = 1
Maxima and Minima(aka extrema) Lowest point on the graph Highest point on the graph In this function, the minimum is at y = 1 In this function, the minimum is at y = -2
Intervals of Increase and Decrease • By looking at the graph of a piecewise function, we can also see where its slope is increasing (uphill), where its slope is decreasing (downhill) and where it is constant (straight line). We use the domain to define the ‘interval’. This function is decreasing on the interval x < -2, is Increasing on the interval -2 < x < 1, and constant over x > 1
Classwork/Homework • Characteristics WS